The Gram dimension gd(G) of a graph is the smallest integer k ≥ 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in Rk, having the same inner products on the edges of the graph. The class of graphs satisfying gd(G) ≤ k is minor closed for fixed k, so it can characterized by a finite list of forbidden minors. For k ≤ 3, the only forbidden minor is Kk+1. We show that a graph has Gram dimension at most 4 if and only if it does not have K5 and K2,2,2 as minors. We also show some close connections to the notion of d-realizability of graphs. In particular, our result implies the characterization of 3-realizable graphs of Belk and Connelly.
Additional Metadata
Keywords matrix completion, semidefinite programming, graph realization, Gram dimension, Euclidean embedding
THEME Logistics (theme 3)
Publisher Springer
Editor A.R. Mahjoub , V. Markakis (Vangelis) , I. Millis , V. Paschos
Series Lecture Notes in Computer Science
Project Semidefinite programming and combinatorial optimization
Conference International Symposium on Combinatorial Optimization
Citation
Laurent, M, & Varvitsiotis, A. (2012). The Gram dimension of a graph. In A.R Mahjoub, V Markakis, I Millis, & V Paschos (Eds.), Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO 2012) (pp. 356–367). Springer.